A two-dimensional delta symbol method and its application to pairs of quadratic forms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910174886756352 |
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| author | Li, Junxian Myerson, Simon L. Rydin Vishe, Pankaj |
| author_facet | Li, Junxian Myerson, Simon L. Rydin Vishe, Pankaj |
| contents | We present a two-dimensional delta symbol method that facilitates a version of the Kloosterman refinement of the circle method, addressing a question posed by Heath-Brown. As an application, we establish the asymptotic formula for the number of integral points on a non-singular intersection of two integral quadratic forms with at least $10$ variables. Assuming the Generalized Lindelöf Hypothesis, we reduce the number of variables to $9$ by performing a double Kloosterman refinement. A heuristic argument suggests our two-dimensional delta symbol will typically outperform known expressions of this type by an increasing margin as the number of variables grows. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_11355 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A two-dimensional delta symbol method and its application to pairs of quadratic forms Li, Junxian Myerson, Simon L. Rydin Vishe, Pankaj Number Theory 11P55 (11D45, 14G05, 14J45, 11D09) We present a two-dimensional delta symbol method that facilitates a version of the Kloosterman refinement of the circle method, addressing a question posed by Heath-Brown. As an application, we establish the asymptotic formula for the number of integral points on a non-singular intersection of two integral quadratic forms with at least $10$ variables. Assuming the Generalized Lindelöf Hypothesis, we reduce the number of variables to $9$ by performing a double Kloosterman refinement. A heuristic argument suggests our two-dimensional delta symbol will typically outperform known expressions of this type by an increasing margin as the number of variables grows. |
| title | A two-dimensional delta symbol method and its application to pairs of quadratic forms |
| topic | Number Theory 11P55 (11D45, 14G05, 14J45, 11D09) |
| url | https://arxiv.org/abs/2411.11355 |